776 research outputs found
The enriched Crouzeix-Raviart elements are equivalent to the Raviart-Thomas elements
For both the Poisson model problem and the Stokes problem in any dimension,
this paper proves that the enriched Crouzeix-Raviart elements are actually
identical to the first order Raviart-Thomas elements in the sense that they
produce the same discrete stresses. This result improves the previous result in
literature which, for two dimensions, states that the piecewise constant
projection of the stress by the first order Raviart-Thomas element is equal to
that by the Crouzeix-Raviart element. For the eigenvalue problem of Laplace
operator, this paper proves that the error of the enriched Crouzeix-Raviart
element is equivalent to that of the Raviart-Thomas element up to higher order
terms
Fully computable a posteriori error bounds for eigenfunctions
Fully computable a posteriori error estimates for eigenfunctions of compact
self-adjoint operators in Hilbert spaces are derived. The problem of
ill-conditioning of eigenfunctions in case of tight clusters and multiple
eigenvalues is solved by estimating the directed distance between the spaces of
exact and approximate eigenfunctions. Derived upper bounds apply to various
types of eigenvalue problems, e.g. to the (generalized) matrix, Laplace, and
Steklov eigenvalue problems. These bounds are suitable for arbitrary conforming
approximations of eigenfunctions, and they are fully computable in terms of
approximate eigenfunctions and two-sided bounds of eigenvalues. Numerical
examples illustrate the efficiency of the derived error bounds for
eigenfunctions.Comment: 27 pages, 8 tables, 9 figure
Two-sided bounds for eigenvalues of differential operators with applications to Friedrichs', Poincar\'e, trace, and similar constants
We present a general numerical method for computing guaranteed two-sided
bounds for principal eigenvalues of symmetric linear elliptic differential
operators. The approach is based on the Galerkin method, on the method of a
priori-a posteriori inequalities, and on a complementarity technique. The
two-sided bounds are formulated in a general Hilbert space setting and as a
byproduct we prove an abstract inequality of Friedrichs'-Poincar\'e type. The
abstract results are then applied to Friedrichs', Poincar\'e, and trace
inequalities and fully computable two-sided bounds on the optimal constants in
these inequalities are obtained. Accuracy of the method is illustrated on
numerical examples.Comment: Extended numerical experiments and minor corrections of the previous
version. This version has been accepted for publication by SIAM J. Numer.
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